The beets and carrots can be prepared in advance and incorporated with the rest of the ingredients just before serving.
Place beets in a medium-sized saucepan with the vinegar and a large pinch of salt. Fill with water to cover. Cover with tin foil. Poke 4 holes in foil. Cook until beets are tender (a knife can easily pierce through), about 45 minutes. Allow beets to cool, then cut each into 8 wedges. Chill in refrigerator.
Pour a tablespoon of oil into a hot saute pan. Add the carrots and cook until golden brown on one side (medium-high heat). Add sugar and stir, cooking until tender and crispy, about 5-7 minutes. Allow carrots to cool a bit, then chill in refrigerator.
Put the watercress in a salad bowl. Add the radishes, beets, and carrots. Season with salt, freshly ground pepper, and a drizzle of olive oil.